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Theorem equvini 2084
Description: A variable introduction law for equality. Lemma 15 of [Monk2] p. 109, however we do not require  z to be distinct from  x and  y (making the proof longer). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 7-Apr-2018.)
Assertion
Ref Expression
equvini  |-  ( x  =  y  ->  E. z
( x  =  z  /\  z  =  y ) )

Proof of Theorem equvini
StepHypRef Expression
1 equcomi 1692 . . . . . 6  |-  ( z  =  x  ->  x  =  z )
21alimi 1569 . . . . 5  |-  ( A. z  z  =  x  ->  A. z  x  =  z )
3 a9e 1953 . . . . 5  |-  E. z 
z  =  y
42, 3jctir 526 . . . 4  |-  ( A. z  z  =  x  ->  ( A. z  x  =  z  /\  E. z  z  =  y
) )
5 19.29 1607 . . . 4  |-  ( ( A. z  x  =  z  /\  E. z 
z  =  y )  ->  E. z ( x  =  z  /\  z  =  y ) )
64, 5syl 16 . . 3  |-  ( A. z  z  =  x  ->  E. z ( x  =  z  /\  z  =  y ) )
76adantl 454 . 2  |-  ( ( x  =  y  /\  A. z  z  =  x )  ->  E. z
( x  =  z  /\  z  =  y ) )
8 a9e 1953 . . . . . 6  |-  E. z 
z  =  x
98, 1eximii 1588 . . . . 5  |-  E. z  x  =  z
109jctl 527 . . . 4  |-  ( A. z  z  =  y  ->  ( E. z  x  =  z  /\  A. z  z  =  y
) )
11 19.29r 1608 . . . 4  |-  ( ( E. z  x  =  z  /\  A. z 
z  =  y )  ->  E. z ( x  =  z  /\  z  =  y ) )
1210, 11syl 16 . . 3  |-  ( A. z  z  =  y  ->  E. z ( x  =  z  /\  z  =  y ) )
1312adantl 454 . 2  |-  ( ( x  =  y  /\  A. z  z  =  y )  ->  E. z
( x  =  z  /\  z  =  y ) )
14 nfeqf 2010 . . . 4  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  F/ z  x  =  y )
15 ax-8 1688 . . . . . 6  |-  ( x  =  z  ->  (
x  =  y  -> 
z  =  y ) )
1615anc2li 542 . . . . 5  |-  ( x  =  z  ->  (
x  =  y  -> 
( x  =  z  /\  z  =  y ) ) )
1716equcoms 1694 . . . 4  |-  ( z  =  x  ->  (
x  =  y  -> 
( x  =  z  /\  z  =  y ) ) )
1814, 17spimed 1961 . . 3  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( x  =  y  ->  E. z
( x  =  z  /\  z  =  y ) ) )
1918impcom 421 . 2  |-  ( ( x  =  y  /\  ( -.  A. z 
z  =  x  /\  -.  A. z  z  =  y ) )  ->  E. z ( x  =  z  /\  z  =  y ) )
207, 13, 19pm2.61ddan 769 1  |-  ( x  =  y  ->  E. z
( x  =  z  /\  z  =  y ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360   A.wal 1550   E.wex 1551
This theorem is referenced by:  equvin  2088  sbequi  2112  sbequiOLD  2116
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-nf 1555
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